An algebraic approach to laying a ghost to rest

TL;DR

This paper presents an algebraic method to reformulate the Pais-Uhlenbeck fourth-order field theory, eliminating ghosts and making it compatible with Dirac's quantum framework, thus advancing its physical acceptability.

Contribution

It introduces a Hamiltonian construction that renders the Pais-Uhlenbeck model ghost-free within Dirac's theory using reduction of order techniques.

Findings

A Hamiltonian free of ghosts for the Pais-Uhlenbeck model is constructed.

The reduction of order method successfully transforms the fourth-order equation into a second-order system.

The approach aligns the model with physical quantum principles without abandoning Dirac's framework.

Abstract

In the recent literature there has been a resurgence of interest in the fourth-order field-theoretic model of Pais-Uhlenbeck \cite {Pais-Uhlenbeck 50 a}, which has not had a good reception over the last half century due to the existence of {\em ghosts} in the properties of the quantum mechanical solution. Bender and Mannheim \cite{Bender 08 a} were successful in persuading the corresponding quantum operator to `give up the ghost'. Their success had the advantage of making the model of Pais-Uhlenbeck acceptable to the physical community and in the process added further credit to the cause of advancement of the use of ${\cal PT} $ symmetry. We present a case for the acceptance of the Pais-Uhlenbeck model in the context of Dirac's theory by providing an Hamiltonian which is not quantum mechanically haunted. The essential point is the manner in which a fourth-order equation is rendered into a system of second-order equations. We show by means of the method of reduction of order \cite {Nucci} that it is possible to construct an Hamiltonian which gives rise to a satisfactory quantal description without having to abandon Dirac.